richardson_extrap

Estimate order of convergence of a response as model fidelity increases

Specification

• Alias: None

• Arguments: None

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Required (Choose One)	Verification Mode	estimate_order	Compute the best estimate of the convergence order from three points
		converge_order	Refine until the estimated covergence order converges
		converge_qoi	Refine until the response converges
Optional		refinement_rate	Rate at which the state variables are refined
Optional		convergence_tolerance	Stopping criterion based on objective function or statistics convergence
Optional		max_iterations	Number of iterations allowed for optimizers and adaptive UQ methods
Optional		model_pointer	Identifier for model block to be used by a method

Description

Solution verification procedures estimate the order of convergence of the simulation response data during the course of a refinement study. This branch of methods is new and currently only contains one algorithm: Richardson extrapolation.

Refinement of the model

The model fidelity must be parameterized by one or more continuous state variable(s).

The refinement path is determined from the initial_state of the continuous_state variables specification in combination with the refinement_rate, where each of the state variables is treated as an independent refinement factor and each of the initial state values is repeatedly divided by the refinement rate value to define new discretization states.

Results

Three algorithm options are currently provided:

• estimate_order

• converge_order

• converge_qoi

Stopping Criteria

The method employs the max_iterations and convergence_tolerance method independent controls as stopping criteria.

Theory

In each of these cases, convergence order for a response quantity of interest (QoI) is estimated from

\[p = ln\left(\frac{QoI_3 - QoI_2}{QoI_2 - QoI_1}\right)/ln(r)\]

where \(r\) is the uniform refinement rate specified by refinement_rate.